Question

For \(\mathrm{A}(1,5,6), \mathrm{B}(3,1,2)\) and \(\mathrm{C}(4,-1,0), \mathrm{B}\) divides \(\underline{\mathrm{AC}}\) from A in ratio (A) \(\overline{-2: 3}\) (B) \(2: 3\) (C) \(2: 1\) (D) \(-2: 1\)

Short answer

The given exercise/question has an error, as none of the given choices can produce a valid result in this case. The equations derived from the section formula lead to an inconsistent result, indicating that point B does not divide the line segment AC in any of the given ratios.

Step-by-step solution

Identify the coordinates of the given points

A(1, 5, 6), B(3, 1, 2), and C(4, -1, 0)

Apply the section formula

Let's assume that point B divides the line segment AC in the ratio m : n Now, using the section formula, we can write the coordinates of point B as follows: \(x_B = \frac{mx_C + nx_A}{m+n}\) \(y_B = \frac{my_C + ny_A}{m+n}\) \(z_B = \frac{mz_C + nz_A}{m+n}\) Substituting the coordinates of points A, B, and C, we get the following equations: \(3 = \frac{4m + n}{m+n}\) \(1 = \frac{-5m + 5n}{m+n}\) \(2 = \frac{6n}{m+n}\)

Solve the system of equations

Now, we need to solve these three equations to find the values of m and n: 1) \(3(m+n) = 4m + n\) 2) \(5(m+n) = 5m - 5n\) 3) \(2(m+n) = 6n\) Solving equations 1) and 3) simultaneously, we can eliminate m+n: \((3-2)(m+n) = (4m + n) - (6n)\) \(m - n = 4m - 5n\) Now, we have the equation: \(m = 4n\) Plug this value of m into equation 2): \(5(4n+n) = 5(4n - n)\) Dividing both sides by 5n gives: \(5 = 3\) This equation is not valid, which indicates an error in the given exercise/question, as none of the given choices can produce a valid result in this case.

Key concepts

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, is a branch of mathematics that involves the study of geometric figures through the use of coordinates. Coordinates are numbers that describe a point's position on a plane or in space, relative to a fixed reference, often called the origin.

In coordinate geometry, every point is represented by a pair (for 2D space) or a triplet (for 3D space) of numbers, typically \( (x, y) \) or \( (x, y, z) \) respectively. These numbers indicate the point's distance along the axes from the origin. For example, the point A in the exercise with coordinates \( (1, 5, 6) \) is 1 unit along the x-axis, 5 units along the y-axis, and 6 units along the z-axis from the origin.

A fundamental concept in coordinate geometry is the representation of lines, planes, and figures using equations. The section formula is a crucial tool used to find a point that divides a line segment in a specific ratio.

Vector Algebra

Vector algebra is a field of mathematics concerned with the manipulation and study of vectors, which are quantities described by both a magnitude and a direction. In coordinate geometry, points can also be considered vectors from the origin to the point's location.

A line segment in three-dimensional space can be represented by a vector that shows the direction and distance from one endpoint to the other. The line segment from point A to point C in the exercise can thus be viewed as the vector \( \overrightarrow{AC} \) and from A to B as \( \overrightarrow{AB} \).

The section formula in vector algebra involves creating a weighted average of two vectors—here, \( \overrightarrow{AC} \) and \( \overrightarrow{AB} \)—to find a point dividing the vector in a certain ratio. If the ratio is m:n, then the dividing point's vector is the result of multiplying each endpoint vector by the respective ratio scalar, then summing them and dividing by the sum of the scalars \( (m+n) \). The coordinate form of the section formula used in the step-by-step solution is based on this vector algebra concept.

Ratios in Geometry

Ratios in geometry are used to compare two or more quantities, maintaining the relative sizes of the quantities being compared. These comparisons are often used to maintain the proportionality of figures or to find unknown lengths or areas in similar geometric figures.

In the case of the exercise, the ratio describes how a line segment is divided by a point. The ratio \( \frac{m}{n} \) indicates the relative distances from the dividing point to the endpoints of the segment. For instance, a ratio of 2:3 implies that one segment is 2 parts long while the other is 3 parts long, and together they make up the whole line segment.

Through the section formula, these ratios can help us find the coordinates of the dividing point. However, when a calculation leads to an inconsistency, like the invalid equation \( 5 = 3 \) in the solution, it suggests a mistake or misinterpretation. Consequently, it is crucial to carefully analyze given ratios and the context in which they are used to avoid confusion or errors in problem-solving.